Chabauty Limits of Fermat Spirals

TL;DR

This work analyzes Chabauty limits of Fermat spirals, proving that all non-empty limits are translations of closed subgroups of $\mathbb{R}^2$ and that, when $\alpha$ is badly-approximable, these limits are lattices of co-volume $\pi$ with an explicit parametrisation from continued fraction convergents of $\alpha$. The authors provide a detailed proof that such limits arise as $X$ is translated and rescaled along sequences, and they classify the possible lattices up to rotation, also establishing that no Fermat spiral is a dense forest. The results connect Diophantine approximation with geometric limit structures in the plane and yield concrete implications for density and visibility questions related to the Danzer problem. Open questions remain for irrational $\alpha$ not badly-approximable and for whether every lattice with co-volume $\pi$ occurs as a limit.

Abstract

A Fermat spiral is a set of points of the form $\sqrt{n}e^{2πiαn}$ for $α\in \mathbb{R}$. In this paper we prove that the Chabauty limits of Fermat spirals are always closed subgroups of $\mathbb{R}^2$, and conclude that no Fermat spirals are dense forests. Furthermore, we show that if $α$ is badly approximable the Chabauty limits are always lattices, for which we give a characterisation.

Theorems & Definitions (30)

• Definition
• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Definition
• Proposition 4
• proof
• Corollary 5
• proof